When dealing with systems which are made of a large number of particles, one is often interested in the collective behaviour of the system rather than in a detailed description. Established approaches in statistical mechanics and kinetic theory allow one to study the limit as the number of particles N tends to infinity and to obtain a (low dimensional) PDE for the evolution of the density of particles. The limiting PDE is a non-linear equation, where the non-linearity has a specific structure and is called a McKean-Vlasov nonlinearity. Of course one of the issues here is which properties of the initial particle systems are preserved upon passing to the limit – and this is something we will touch upon. Even if the particles evolve according to a stochastic differential equation, the limiting equation is deterministic, as long as the particles are subject to independent sources of noise. If the particles are subject to the same noise (common noise) then the limit is given by a Stochastic Partial Differential Equation (SPDE). In the latter case the limiting SPDE is substantially the McKean-Vlasov PDE + noise; noise is further more multiplicative and has gradient structure. One may then ask the question about whether it is possible to obtain McKean-Vlasov SPDEs with additive noise from particle systems. We will explain how to address this question, by studying limits of weighted particle systems. We will moreover discuss applications of the problem of sampling from the invariant distribution of SPDEs with additive noise.
Interacting particle systems, McKean-Vlasov PDEs and S(P)DEs with additive noise
.pdf [725Kb]
